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Alg II Unit 4-1 Quadratic Functions and Transformations


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Alg II Unit 4-1 Quadratic Functions and Transformations

  1. 1. 4-1 QUADRATIC FUNCTIONS ANDTRANSFORMATIONSChapter 4 Quadratic Functions and Equations©Tentinger
  2. 2. ESSENTIAL UNDERSTANDING ANDOBJECTIVES Essential Understanding: The graph of any quadratic function is the transformation of the graph of the parent function y = x2 Objectives: Students will be able to:  Identify and graph quadratic functions  Identify and graph the transformations of quadratic functions (reflect, stretch, compression, translation)  Solve for the minimum and maximum values of parabolas
  3. 3. IOWA CORE CURRICULUM Algebra A.CED.1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Functions F.IF.4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. F.IF.6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. F.IF.7. Graph functions expressed symbolically, and show features of the graph, by hand in simple cases and using technology for more complicated cases. F.BF.3. Identify the effect on the graph of f(x) + k, kf(x), f(kx), and f(x+k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.
  4. 4. VOCABULARY Parabola: the graph of a quadratic function, it makes a U shape Quadratic Function: ax2 + bx + c Vertex Form: f(x) = a(x – h)2 +k, where a doesn’t equal zero, vertex is (h, k) Axis of Symmetry: line that divides the parabola into two mirror images. Equation x = h Parent Function: y = x2
  6. 6. GRAPHING A QUADRATIC FUNCTION Graphing a Function in the form f(x) = ax2 f(x) = (1/2)x2 Plot the vertex Find and plot two points on one side of the axis of symmetry Plot the corresponding points on other side of the axis of symmetry Sketch the curve Graph: f(x) = -(1/3)x2 What can you say about the graph of the function f(x) = ax2 if a is a negative number?
  7. 7. TRANSFORMATIONS Vertex form: f(x) = a(x-h)2 + k Reflection: if a is positive the graph opens up, if a is negative it reflects across the x-axis and opens downward If the parabola opens upward, the y coordinate of the vertex is a minimum If the parabola opens downward, the y coordinate of the vertex is a maximum Stretch a > 1 the graph becomes more narrow Compression 0< a < 1 the graph becomes more flat
  8. 8. TRANSFORMATIONS Standard form: f(x) = a(x-h)2 + k Vertical Translation: k value, on the outside of the parentheses. Moves graph up and down Horizontal translation: opposite of the h value, on the inside of the parentheses. Moves graph left and right.
  9. 9. EXAMPLES For the equations below, write the vertex, the axis of symmetry, the max or min value, and the domain and range. Then describe the transformations. f(x) = x2 – 5 f(x) = (x – 4)2 f(x) = -(x + 1)2 f(x) = 3(x – 4)2 – 2 f(x) = -2(x +1)2
  10. 10. HOMEWORK Pg. 199 – 200 # 9-33 (3s) 35-37, 38, 40 – 42